Questions on the mathematics behind the famed toy invented by Ernő Rubik.

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Rubik's cube computing algorithm

Is it possible to solve a 3x3 Rubik's cube using a tree similar to a 'mini-max game tree' (listing all possible unique moves, then listing further moves from this and so on), or is the sample space ...
2
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0answers
90 views

Minimum number of stickers required for 3x3 Rubik's Cube

Lately, the stickers on my V-Cube have been peeling off, and I became curious: what is the minimum number of stickers I would actually need in order to represent each unique configuration? I have but ...
6
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1answer
277 views

Miniature Rubik's Cube

$R$ is a clock-wise twist of a right-hand face and $U$ is a clock-wise twist of the upper face of a $2 \times 2 \times 2$ Rubik's Cube. Find the order of the subgroup generated by $\{R, U\}$
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1answer
50 views

How to use group theory to solve larrys square iphone app

There's a 2 d version of rubiks cube on apple app store. How can group theory give an algorithm to solve the iPhone app:larry's square.
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2answers
127 views

Is there a proof, that for any Rubik's cube, after solving the first 2 layers, there will always be an even amount of oriented edges?

I've always wondered if there was a proof for this, since the beginners method to solving the last layer depends on this (you either have no edges oriented correctly, an L, a line or a cross, and each ...
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1answer
121 views

Complex automata Rubiks cube question (with picture) help needed

Question 1 A Rubik’s Cube is a puzzle in the shape of a cube. Each face is covered by nine stickers, each of which is coloured with one of six colours: white, red, blue, orange, green, andyellow. An ...
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1answer
100 views

Number of unique permutations of a 3x3x3 cube

Given a 3x3x3 cube (like a rubik's cube) where each of the 27 cubes has a distinct number, how many unique permutations are possible? Simple rotations of the entire cube should not be counted. The ...
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43 views

identifying a subgroup of $S_8$ generated by 4-cycles

Let $G \subseteq S_8$ be the subgroup generated by some 4-cycles. If we number the elements $1,2,\dots, 8$, the 4-cycles are $(1234),(5678),(1485),(2376),(1265),(4378)$ I am not sure if I have ...
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1answer
247 views

Creating a Rubik's Cube Algorithm

So, I realize this isn't a very math-oriented question, but I thought it fit better here than anywhere else. Anyway, I was simply wondering how people come up with the many Rubik's Cube algorithms, ...
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0answers
40 views

Rubik's cube with different neighboring color

How many valid configuration of the 3x3 Rubik's cube are there so that no two adjacent faces are the same color? Thanks.
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4answers
192 views

Minimum number of random moves needed to uniformly scramble a Rubik's cube?

Follow up from my last question: $3 \times 3$ Rubik's cube scrambling question I am talking about $3 \times 3$ Rubik's cubes. Start with a solved cube. Then make some amount of random moves ...
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3answers
102 views

$3 \times 3$ Rubik's cube scrambling question

It was proven that a Rubik's cube needs at most $20$ moves to solve. This implies that any configuration of a Rubik's cube can be reached from an unscrambled Cube in at most $20$ moves. So, say when ...
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1answer
52 views

Eulerian path for Rubik's Cube states

There are a number of discussions online confirming that there exists a Hamiltonian cycle through the states of a Rubik's Cube. Or more precisely, the "quarter-turn metric Cayley graph for the Rubik's ...
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2answers
213 views

Can RUBIK's cube be solved using group theory?

Can RUBIK's cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?
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1answer
54 views

Independently rotating cubelets of a 2 x 2 rubiks cube?

I was reading up on 2 x 2 Rubik's cube. Wikipedia says that Any permutation of the eight corners is possible (8! positions), and seven of them can be independently rotated (37 positions). I ...
4
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1answer
60 views

What is “tetrad twist” in relation to Rubik's cubes?

I am trying to write a computer program to solve Rubik's cubes using a version of Thistlethwaite's algorithm. Instead of using lookup tables to perform each phase of the algorithm, I'm using a ...
33
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3answers
3k views

Can a Rubik's cube be mapped knowing only two sides?

Is it possible to know the entire configuration of a Rubik's cube looking at only two sides and not rotating the cube? In other words: what is the minimum information required to create a ...
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1answer
84 views

What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;] Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn) ...
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2answers
66 views

Rubik cube number of alternative solutions

If a cube is in a configuration that requires 20 moves to solve, is that sequence unique, or are there multiple sequences that arrive at a solution? That is: are there are two or more sequences that ...
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2answers
453 views

Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?

I'm not sure if this question belongs here but I see lots of Rubik Cube's questions around so here it goes: Can I take a standard 3x3 Rubik's Cube and shuffle it so that, for every face, there are no ...
2
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4answers
252 views

Unreachable rubik cube positions.

Are there positions of the rubik cube which cannot be reached by applying the standard moves starting from the solved cube? If so, how many such positions are there?
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1answer
43 views

Rubik's cube question showing surjectivity

Suppose $f: G \rightarrow S_C$ be the action of Rubik's group G on the collection $C$ of corner cubelets. I want to show that the action $f$ is onto $S_C$.
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1answer
105 views

Rubik's Group Question - Commutators

Suppose that $f: G \rightarrow S_C$ be the action of Rubik's group $G$ on the collection of $C$ of corner cubelets. So if $C_1$ and $C_2$ be distinct corner cubelets. Show that there is a move $M \in ...
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2answers
119 views

What are the total number of distinguishable configurations of a rubik's cube?

Given a standard $3 \times 3 \times 3$ rubik's cube, what are the total number of distinguishable configurations of the cube? Two configurations are called distinguishable iff one can be rotated to ...
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0answers
52 views

How to “validate” a Rubik's Cube configuration? [duplicate]

Looking at a picture of a Rubik's cube how can I tell if it's in one of its possible configurations (image is not fake)? By "possible configuration" I mean a configuration that can be achieved by ...
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0answers
147 views

Faces on a rotating cube

This one is probably a softball question, but... Fact: A cube has 6 faces Fact (unless my math is wrong): A cube that is cut in half center horizontal can be reoriented to have 18 unique faces. ...
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2answers
409 views

combinations of Rubik cube

A Rubick's Cube has owl heads on it, which can be misoriented. How many (times) MORE combinations are there of this cube vs. one that has blank stickers? Can someone give me some hints? Thanks
2
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1answer
296 views

Rubik's cube and counting

Inspired by this question, I formulate the following: Suppose I have a $3\times3\times3$ Rubik's cube, call each small square on the surface a piece, there are then $3*3*6 = 54$ pieces. Enumerate ...
8
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1answer
459 views

Literature on group theory of Rubik's Cube

While searching for literature on the group theory of Rubik's Cube, I mostly find introductions to group theory motivated by applications to Rubik's cube. I.e. the focus lies on elementary group ...
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3answers
1k views

Why should Rubik's cube get attention from mathematicians?

I've seen a lot of math debate, calculations and other stuff related to Rubik's cube lately, but I don't really understand why is it important, why should anyone spend time asking and answering ...
14
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2answers
623 views

Rubik Cube finite non-abelian group

I was reading the following paper from MIT: http://web.mit.edu/sp.268/www/rubik.pdf The paper is not difficult to understand, it is more or less a short introduction into group theory, taking the ...
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1answer
157 views

Rubik's cube as $2\times2\times2$ or $2\times2$? [closed]

I'm surprised why people call a Rubik's cube as $2 \times2\times2$ or $3\times3\times3$? It's a CUBE! Isn't saying $2\times2$ or $3\times3$ cube (or rather just $2$ or $3$) sufficient?
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2answers
171 views

Lower bounds on numbers of arrangements of a Rubik's cube

Last night, a friend of mine informed me that there were forty-three quintillion positions that a Rubik's Cube could be in and asked me how many there were for my Professor's Cube (5x5x5). So I gave ...
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4answers
2k views

Minimum number of steps needed to solve a rubik cube

Long time ago I've seen a book on group theory and there was an appendix about rubik cube. I remember there were only three steps that enabled me to solve my cube (three strings with letters encoding ...
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1answer
132 views

The redundancy of Rubik's cube states [duplicate]

Possible Duplicate: Rubik’s Cube Not a Group? I take a Rubik's cube in the solved state, and I secretly assign a unique integer label to each of the cubies. I then, via an arbitrarily long ...
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2answers
423 views

Rubik's Cube Combination

Could anyone explain why the number of legal or reachable combinations of a $3\times 3\times 3$ Rubik's Cube is $1/12\mbox{th}$ of the total. I understood the logic behind the total number of ...
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3answers
555 views

Presentation of Rubik's Cube group

The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
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3answers
2k views

Brute force method of solving the cube: How many moves would it take?

Given that Rubik's cube has finitely many positions, one possible "brute force" method to solve it would be to determine once a sequence of moves which eventually reaches every possible position of ...
8
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2answers
705 views

Orders of moves in the Rubik's cube

Playing with my Rubik's cube, I was thinking of facts about it that are immediate to mathematicians but novel to others. Here's one: Given a Rubik's cube in the solved state, any sequence of moves, ...
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1answer
135 views

Rubik's cube notation [closed]

Various Rubik cube guides use the notation F and R to describe the front and right faces of the cube. However, in the illustrations they show the cube in isometric projection. So which side is right, ...
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3answers
2k views

Is there a “most random” state in Rubik's cube?

Is there a state in Rubik's cube which can be considered to have the highest degree of randomness (maximum entropy?) asssuming that the solved Rubik's cube has the lowest?
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1answer
226 views

Are actions in the $3\times 3\times 3$ rubik cube a group?

Are actions in the $3\times 3\times 3$ rubik cube a group? You can see here Rubik's Cube Not a Group? that $4\times 4\times 4$ rubik cubes or higher arent groups. But what about $3\times 3\times ...
21
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1answer
1k views

Rubik's Cube Not a Group?

I read online that although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all. How can that be true? There is obviously an identity and it is closed, so ...
3
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2answers
518 views

Rubik's cube puzzle

If we cut along the plane orthogonal to the largest diagonal of a Rubik's cube, what is the maximum number of small cubes can we cut? I thought this should be $9$, but apparently this is not the ...
20
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2answers
6k views

How to tell if a Rubik's cube is solvable

How can I determine if a certain Rubik's cube, that is in a certain state, is solveable? By "certain state" it could mean that the cube has been dismantled and put together again. And in my experience ...
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2answers
740 views

All possible permutations on a Rubik cube ($3\times3\times 3$) can be reached from the initial state?

If I were to represent a state in the Rubik cube as a permutation of the colors on the 9 tiles per side on all sides of the cube, could I reach all possible states (i.e. colorings) by the permutations ...
3
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1answer
343 views

Does solving a Rubik's cube imply alignment?

Today, I got my hands on a Rubik's cube with text on it. It looks like this: Now, I would love to know whether solving the right cube will also always correctly align the text on the cube or ...
13
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2answers
557 views

Mathematics Behind the 4×4 and 5×5 Rubik's Cube

A lot is known about the math behind the 3×3 Rubik's cube (symmetries, generators, group structure etc...). Is the same true for the 4×4 and 5×5 cubes? I haven't had much success finding this ...
6
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1answer
544 views

Solving a scrambled $3 \times 3 \times 3$ Rubik's Cube with at most 20 moves!

I read somewhere that any scrambled form of $3 \times 3 \times 3$ Rubik's cube can be solved using at most $20$ moves, and I just said "wow"! I am wondering can we prove this by mathematical ways? Or ...
10
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1answer
2k views

What are the 2125922464947725402112000 symmetries of a Rubik's Cube?

In a recent talk, Marcus du Sautoy says there are 2125922464947725402112000 (2.1*10^24) symmetries of a Rubik's cube, but doesn't explicitly identify what qualifies as a symmetry. What counts as a ...